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Unformatted text preview: Graph Coloring Margaret M. Fleck 3 May 2010 This lecture discusses the graph coloring problem (section 9.8 of Rosen). 1 Announcements Makeup quiz last day of classes (at the start of class). Your room for the final exam (Friday the 7th, 710pm) is based on the first letter of your last name: AH Roger Adams Lab 116 IW Business Instructional Facility (515 E. Gregory opposite Armory) 1001 XZ Roger Adams Lab 116 The conflict exam (Monday the 10th, 1:304:30pm) is in 269 Everett Lab. 2 The problem Were only going to talk about simple undirected graphs. Remember that two vertices are adjacent if they are directly connected by an edge. 1 A coloring of a graph G assigns a color to each vertex of G , with the restriction that two adjacent vertices never have the same color. The chro matic number of G , written ( G ), is the smallest number of colors needed to color G . For example, only three colors are required for this graph: R G G B R But K 4 requires 4: R G B Y 3 Computing colorings For certain classes of graphs, we can easily compute the chromatic number. For example, the chromatic number of K n is n , for any n . Notice that we have to argue two separate things to establish that this is its chromatic number: K n can be colored with n colors. K n cannot be colored with less than n colors For K n , both of these facts are fairly obvious. Assigning a different color to each vertex will always result in a wellformed coloring (though it may be 2 a waste of colors). Since each vertex in K n is adjacent to every other vertex, no two can share a color. So fewer than n colors cant possibly work. Similar, the chromatic number for K n,m is 2. We color one side of the graph with one color and the other side with a second color. In general, however, coloring requires exponential time. Theres a couple specific versions of the theoretical problem. I could give you a graph and ask you for its chromatic number. Or I could give you a graph and an integer k and ask whether k colors is enough to be able to color G . These problems are all NPcomplete or NPhard. That is, they apparently require exponential time to compute, even though in some cases its easy to check that their output is correct. (They are exponential if the P and NP classes of algorithms are actually different, which is one of the big...
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This note was uploaded on 12/24/2010 for the course CS CS 173 taught by Professor Fleck during the Spring '10 term at University of Illinois, Urbana Champaign.
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